Probability
Probability
Probability measures how likely an event is on a scale from 0 to 1. You must list sample spaces, count equally likely outcomes, use the complement rule, and combine probabilities for mutually exclusive and independent events. Know that P(not A) = 1 - P(A), that mutually exclusive events cannot occur together, and that independent events do not affect each other. Tree diagrams and two-way tables are useful for multi-stage experiments. Relative frequency from data can estimate probability.
Common traps are forgetting to reduce the denominator in without-replacement problems, adding probabilities for events that can overlap, and confusing independent with mutually exclusive. For two events A and B, P(A or B) = P(A) + P(B) - P(A and B) when they can overlap. For independent events, multiply: P(A and B) = P(A) x P(B). In without-replacement questions, the total number of items and the number of favourable items both decrease after the first draw.
Worked example: a bag contains 3 red and 2 blue counters. Two are taken without replacement. P(first red) = 3/5. After a red is removed, 2 red and 2 blue remain, so P(second red) = 2/4. Thus P(both red) = 3/5 x 2/4 = 6/20 = 3/10.
Rule of thumb: write down the sample space or the changing totals at each stage before multiplying probabilities.